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Unit 6: Radical Expressions and Equations

Roots and rational exponents as the same idea in two notations, and the equation-solving step that manufactures fake answers.

6 topics · 22 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Rational Exponents

  1. How it worksHow does the denominator of a rational exponent become the index of a root?
  2. Worked exampleFind the value of 27 to the power two-thirds without a calculator.
  3. Common mistakeWhy doesn't 8 to the power one-third equal 8 divided by 3?
Your turn1 of 2
Simplify (x3)2/3, for x > 0, as a function of x.

Simplifying Radical Expressions

  1. How it worksHow does factoring out a perfect square let you simplify a radical expression?
  2. Worked exampleFind the simplified form of the square root of 200.
  3. Common mistakeWhy isn't the square root of x squared automatically just x, once negative numbers are allowed?
  4. Going furtherWhat changes about simplifying a radical when its index is 4 instead of 2, like a fourth root?
Your turn1 of 2
Find the simplified form of the square root of (72·7).

Adding, Subtracting, and Multiplying Radicals

  1. How it worksHow does multiplying two radicals combine what's underneath into a single root?
  2. Worked exampleFind the simplified sum of 3 times the square root of 8 and 2 times the square root of 18.
  3. Common mistakeWhy can't the square root of 4 plus the square root of 9 be combined into the square root of 13?
  4. Ties togetherHow is combining like radical terms the same rule as combining like terms in a polynomial?
Your turn1 of 2
Find the simplified sum of 7 times the square root of (92·2) and 7 times the square root of (42·2).

Rationalizing the Denominator

  1. How it worksHow does multiplying by a clever form of 1 clear a square root out of a denominator?
  2. Worked exampleFind the rationalized form of 5 over the square root of 3.
  3. Common mistakeWhy does rationalizing a binomial denominator like 2 plus the square root of 5 need its conjugate?
Your turn1 of 2
Find the rationalized form of 7 over the square root of 7.

Solving Radical Equations

  1. How it worksHow does squaring both sides of an equation get rid of a square root symbol?
  2. Worked exampleSolve the square root of (2x + 3) equals 5 for x.
  3. Common mistakeWhy isn't checking your solution optional once you've squared both sides of an equation?
Your turn1 of 2
Solve the square root of (4x + 6) = 9 for x.

Extraneous Solutions from Squaring

  1. How it worksWatch how squaring x = -3 into x2 = 9 manufactures the extra solution x = 3 out of nowhere.
  2. Worked exampleSolve the square root of (x + 7) equals x + 1, and show why one of the two answers has to be rejected.
  3. Common mistakeWhy doesn't a negative number under a radical automatically mean there's no answer, now that complex numbers exist?
  4. Common mistakeWhy doesn't plugging a solution back into the squared version of an equation catch what checking the original equation catches?
  5. Ties togetherHow does the extraneous-solution problem in radical equations connect to the same issue in rational equations?
Your turn1 of 2
Solving x = -7 by squaring gives x2 = 72. What is the EXTRANEOUS solution introduced by squaring (the one that does NOT satisfy the original equation x = -7)?